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The specific heat of a trapped Fermi gas: An analytical approach

J. M B Noronha, David J. Toms

Published Dec 31, 2000• 0 citations
Publisher

Abstract

We find an analytical expression for the specific heat of a Fermi gas in a harmonic trap using a semi-classical approximation. Our approximation is valid for k(B)T ≥ hw(x, y, z) and in this range it is shown to be highly accurate. We comment on the semi-classical approximation, presenting an explanation for this high accuracy. (C) 2000 Elsevier Science B.V.

DOI
10.25405/ncl.34598073

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We find an analytical expression for the specific heat of a Fermi gas in a harmonic trap using a semi-classical approximation. Our approximation is valid for k(B)T ≥ hw(x, y, z) and in this range it is shown to be highly accurate. We comment on the semi-classical approximation, presenting an explanation for this high accuracy. (C) 2000 Elsevier Science B.V.
In the Thomas-Fermi semi-classical approximation, the maximal trap range and the real trap volume of ideal Fermi gas in an n-dimensional potential trap are gaven, and the relevant equations of state are derived. These results indicate that the maximal trap range and the real pressure of trapped gas are related to the potential field and the chemical potential of the free and ideal Fermi system. When the Thomas-Fermi approximate is valid and the condition ((kT)/(hω))2 ((16π2g)/(9N))2/3<<1 is satisfied, the application of the equation of state to three-dimensional spherical symmetry harmonic trap yields the result that the change of pressure is not obvious when the temperature changes, but the change of pressure is closely related to mass of particle, number of particles and the frequency of harmonic potential.
We study the thermodynamic properties of an ideal gas of fermions in a harmonic oscillator confining potential. The analogy between this problem and the de Haas-van Alphen effect is discussed and used to obtain analytical results for the chemical potential and specific heat in the case of both isotropic and anisotropic potentials. Step-like behaviour in the chemical potential, first noted in numerical studies, is obtained analytically and shown to result in an oscillatory behaviour of the specific heat when the particle number is varied. The origin of these oscillations is that part of the thermodynamic potential is responsible for the de Haas-van Alphen-type effect. At low temperatures we show analytically that there are significant deviations in the specific heat from the expected linear temperature dependence, again as a consequence of the de Haas-van Alphen part of the thermodynamic potential. Results are given for one, two, and three spatial dimensions. In the anisotropic case we show how the specific heat jumps as the ratio of oscillator frequencies varies. © 2005 Elsevier Inc. All rights reserved.
We study normal state properties of an interacting Fermi gas in an isotropic harmonic trap of arbitrary dimensions. We exactly calculate the first-order perturbation terms in the ground-state energy and chemical potential, and obtain simple analytic expressions of the total energy and chemical potential. At zero temperature, we find that the Thomas–Fermi approximation agrees well with the exact results for any dimension even though the system is small. In the high-temperature (classical) region, we find the interaction energy decreases in proportion to T − d /2 , where T is the temperature and d is the dimension of the system.